Stephen Wolfram and I designed a medal to celebrate Gregory Chaitin’s 60th birthday and his contributions to mathematics. Chaitin is one of the key founders of algorithmic information theory (AIT), which combines, among other elements, Shannon’s information theory and Turing’s theory of computability. He did this independently of Andrei Kolmogorov and Ray Solomonoff when Greg was still a teenager in the mid 1960s.

Among Chaitin contributions are the definition of a random sequence via algorithmic incompressibility, his information-theoretic approach to Gödel’s incompleteness theorem and his halting probability epitomised by his Omega number. His work on Hilbert’s 10th problem has made him believe that in a sense there is randomness even in elementary arithmetic.

The idea of the medal was to somehow replicate the Gottfried Leibniz medallion, an image of which appears at the bottom of Greg’s home page.

Chaitin has spent his career working on foundational questions in mathematics and computation, and in some ways he has been a modernizer of Leibnizian ideas. Leibniz may have been the first computer scientist and information theorist. Early in his life he developed binary arithmetic.

On January 2nd, 1697, Leibniz wrote a letter to Rudolf August, Duke of Braunschweig-Wolfenbüttel, in which he detailed the design of a commemorative coin or medallion which he suggested could be minted in silver. The design he described posited an analogy between “the creation of all from nothing through the omnipotence of God” and the fact that “all numbers [could] be created from zeros and ones”.

So the medal does not commemorate Leibniz’s discovery of binary arithmetic. Rather, his description suggests a medal in which binary arithmetic glorifies God–and the duke. (He proposed that the obverse of the coin bear the Duke’s “face or monogram”). However, Leibniz’s religious ideas are but simple. Newton used to mock him about it, but Leibniz idea of God was way more rational than one would expect from his constant citations to god.

More on the history of Leibniz’ binary language, the letter and the medallion can be found here (pp. 31-36):

[“The binary medallion apparently was never struck*. Numerous writers have based a contrary assumption, in the last analysis, upon having seen some version of its design. The Duke was already 70 years old when he received the medallion proposal in 1697. “(p. 35)

“After a thorough search of the catalogs of applicable coin collections, including all known special Brunswickian collections, Dr. W. Jesse of the Stadtisches Museum Braunschweig reported in his letter of November 2, 1965 that in his opinion, the proposed medallion had never been struck. (p. 51)”

“What actually survives are illustrations in later printings of the letter. Two Versions of Leibniz’s Design of the Binary Medallion. They are facsimiles of the ones appearing on the respective title pages of Johann Bernard Wiedeburg’s Dissertatio mathematica de praestantia arithmeticae binaria prae decimali (Jena: Krebs, 1718) and Rudolf August Nolte’s Leibniz Mathematischer Beweis der Erschaffung und Ordnung der Welt in einem Medallion. Langenheim, 1734. (See pp. 34, 36, 56 for images of the proposed coin, including the obverse side).”]

During the Summer a small group of people from Wolfram Research led by Stephen Wolfram worked together on the design for Chaitin’s 60th birthday medallion. Stephen and I were keen to incorporate representations of the most definitive elements of Chaitin’s influential career as founder of AIT. It was pretty obvious that Chaitin’s medallion had to include the letter Omega representing his Omega number (Chaitin’s Omega gives the halting probability of a (prefix-free) universal Turing machine). We also wanted to show some digits of an Omega number calculated by Cristian Calude, since even though the Omega number is non-computable, Calude managed to calculate an initial segment by using the binary version of Chaitin’s formula following Chaitin’s construction with register machine programs (of course the digits are dependent on the universal Turing machine chosen). The halting and non-halting results for the register machine programs in question were represented by arrows and lines below the letter Omega. Here is the link to Calude’s paper in which he computed the first digits of Chaitin’s Omega number. It includes a section that we used in determining the placement of the arrows in our design:

Cristian S. Calude, Michael J. Dinneen, and Chi-Kou Shu. “Computing a Glimpse of Randomness,” Experimental Mathematics, Vol. 11 (2002), No. 3.

The first 64 bits of Chaitin’s Omega from the paper are:

000000100000010000011000100001101000111111…

0010111011101000010000

However, we decided to use the 40 digits from the standard binary formula version (Chaitin’s original formulation), also calculated by Calude in the same seminal paper:

0001000000010000101001110111000011111010

The upper background of the medallion is a binary circular array conceived by Michael Schreiber and generated with the following code in Mathematica:

Manipulate[Graphics[

{Black, Disk[{0, 0}, p + 2], Table[

Table[{GrayLevel[Mod[a, 2]],

Disk[{0, 0}, q + 1, {2 Pi (a – 1)/(2^q), 2 Pi a/(2^q)}]}, {a, 1, 2^(q),

1}],

{q, p, 1, -1}], White, Disk[]}],

{{p, 3, “bits”}, 1, 8, 1}]

Like Leibniz, we wanted an inscription in timeless Latin, so we began looking for a text to inscribe on Greg’s medallion, one that was related to his seminal work.

One year previously, when I met Chaitin at his office in IBM’s Thomas J. Watson Research Center in Yorktown Heights, New York, he invited me to his home and kindly gave me some of his published books (I already had a couple of them but he completed my collection). In return I sent him a very rare limited edition of a book by Jorge Luis Borges and Alfonso Reyes entitled “La máquina de pensar” (“The thinking machine”). Needless to say I kept a copy for myself! As everybody knows, Borges is a famous Argentinian writer just like Chaitin himself (Chaitin is also American). Reyes is a Mexican writer whom Borges credits as an important influence. Indeed their styles show a degree of similarity. In any case, it turned out that like me, Chaitin liked Borges a lot, but he had never heard of Reyes, whom I happen to like as much as Borges. He told me he had enjoyed the book very much, so some of the first inscriptions proposed for the medal were quotes from Borges from his Babel library. But soon we decided that one of the Leibniz quotations appearing on Chaitin’s webpage would be more appropriate:

*Dieu a choisi celuy qui est… le plus simple en hypotheses et le plus riche en phenomenes.

[God has chosen that which is the most simple in hypotheses and the most rich in phenomena.]

*Mais quand une regle est fort composée, ce qui luy est conforme, passe pour irrégulier.

[But when a rule is extremely complex, that which conforms to it passes for random.]

Greg has suggested that these quotes from Leibniz, among others, are early anticipations of AIT.

But after further discussions with Stephen, we agreed on two of Chaitin’s own most often quoted statements encapsulating his most seminal contributions: “Everything can be summarized in one thing, but that thing cannot be reached” (In other words: All computable facts can be summarized in Chaitin’s Omega number, but that number is not itself computable); and “Mathematical facts are true for no reason” (or by accident, as Chaitin uses to say).

Stephen decided to consult a world expert—a friend of his from high school named Armand d’Angour who is now a Classics professor at Oxford. In 2004 he was commissioned by the International Olympic Committee to compose a Pindaric Ode to Athens which was recited at the Olympic Games. The first thing he pointed out was that Leibniz’s inscription (‘omnibus ex nihilo ducendis sufficit unum’) was a hexameter. D’Angour quickly came up with a pentameter as well for Greg, in his words a “perfect classical one-liner” of the kind that kings in antiquity used to reward poets for. Thus we had a full elegiac couplet, the first line of which read as follows:

Everything can be summarized in one thing, but the thing itself cannot be reached

OMNE UNO IMPLICITUR QUOD NON ATTINGITUR IPSUM.

D’Angour suggested that we replace the “o” in “uno” with an Omega letter (‘Everything can be summarised in one Omega, which itself cannot be attained’).

He added that Latin verse aficionados would enjoy the way the first three words ran into each other, thus demonstrating what the phrase connoted.

The second line which at first read:

Mathematical facts are true by chance

MATHEMATICAE PRINCIPIA FORTUITO VERA

was later turned into the pentametric

FORTUITA EVENIUNT VERA MATHEMATICAE.

The truths of mathematics turn out to be fortuitous.

And beneath this the medal read:

Celebrating the work(s) of Gregory Chaitin MMVII:

AD LAUDEM GC MMVII (where the Leibniz version has IMAGO CREATIONIS INVEN GGL).

D’Angour claims that if he were Greg Chaitin, he would be happy to have all this inscribed on his tombstone. If he were Maecenas, he would consider rewarding the poet with a Sabine Farm.

The Latin inscription on Leibniz’s medallion can be rendered thus: “To make all things from nothing unity suffices” (i.e. You can represent every number using just the digit 1). The inscription on Chaitin’s medallion says: “Everything can be summarized in one [Omega], which cannot itself be attained/The truths of mathematics turn out to be fortuitous”.

Once we had finalized the design, we wondered about the obverse of the medallion. We realized that this was the chance to finally cast Leibniz’ medallion after almost 300 years! So I went about reconstructing it, noting every single detail. I wrote some Mathematica code incorporating all these details which could be used for an electronic design to be finally struck. Here is the Mathematica notebook.

Stephen Wolfram presented the medallion to Chaitin during the NKS Science Conference on the 15th. of July, 2007 at the University of Vermont, Burlington, U.S. The original solid silver medallion was delivered to him on November the 2nd of the same year. Nine more copies were made of Merlin gold, one of which belongs to me (pictures below). The others were given to Chaitin’s relatives, and to Armand D’Angour, Cristian Calude, Jeremy Davis and Stephen Wolfram. Two were retained by WRI’s design department for the archive.